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运筹学学报 ›› 2021, Vol. 25 ›› Issue (3): 1-14.doi: 10.15960/j.cnki.issn.1007-6093.2021.03.001

• •    下一篇

浅谈增广Lagrange方法中的二阶分析

张立卫*   

  1. 大连理工大学数学科学学院, 辽宁大连 116024
  • 收稿日期:2021-03-15 发布日期:2021-09-26
  • 通讯作者: 张立卫 E-mail:lwzhang@dlut.edu.cn
  • 基金资助:
    国家自然科学基金面上项目（No.11971089）

Discussion on second-order analysis in augmented Lagrange method

ZHANG Liwei*   

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China
  • Received:2021-03-15 Published:2021-09-26

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摘要： 从极大化基于增广Lagrange函数的对偶函数的角度，可将增广Lagrange方法的乘子的迭代解释为常步长的梯度方法。增广Lagrange方法的有效性可以通过分析对偶函数的二阶微分得到。给出等式约束优化问题和一般约束非线性规划问题的对偶函数的二阶微分估计，解释为什么常步长的梯度方法具有快的收敛速度。

关键词: 增广Lagrange方法, 对偶问题, 收敛速度, 二阶微分

Abstract: From the point of view of maximizing the dual function based on augmented Lagrange function, the update of multiplier of augmented Lagrange method can be interpreted as a constant step gradient method. The effectiveness of augmented Lagrange method can be obtained by analyzing the second-order differential of dual function. In this paper, the second-order differentials of dual function for equality constrained optimization problem and general constrained nonlinear programming problem are estimated, which explains why the gradient method with constant step size has fast rate of convergence.

Key words: augmented Lagrange method, dual problem, rate of convergence, secondorder differential

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